Bessel Functions - 10.50 Wronskians and Cross-Products

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DLMF Formula Constraints Maple Mathematica Symbolic
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Maple 		Numeric
Mathematica
10.50#Ex1 𝒲 ⁑ { 𝗃 n ⁑ ( z ) , 𝗒 n ⁑ ( z ) } = z - 2 Wronskian spherical-Bessel-J 𝑛 𝑧 spherical-Bessel-Y 𝑛 𝑧 superscript 𝑧 2 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{j}_{n}\left(z\right),% \mathsf{y}_{n}\left(z\right)\right\}=z^{-2}}}
\Wronskian@{\sphBesselJ{n}@{z},\sphBesselY{n}@{z}} = z^{-2}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}} 
Error

Wronskian[{SphericalBesselJ[n, z], SphericalBesselY[n, z]}, z] == (z)^(- 2)
Missing Macro Error Successful - Successful [Tested: 21]
10.50#Ex2 𝒲 ⁑ { 𝗁 n ( 1 ) ⁑ ( z ) , 𝗁 n ( 2 ) ⁑ ( z ) } = - 2 ⁒ i ⁒ z - 2 Wronskian spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑧 spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑧 2 𝑖 superscript 𝑧 2 {\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{h}^{(1)}_{n}}\left(z% \right),{\mathsf{h}^{(2)}_{n}}\left(z\right)\right\}=-2iz^{-2}}}
\Wronskian@{\sphHankelh{1}{n}@{z},\sphHankelh{2}{n}@{z}} = -2iz^{-2}		 

Error

Wronskian[{SphericalHankelH1[n, z], SphericalHankelH2[n, z]}, z] == - 2*I*(z)^(- 2)
Missing Macro Error Successful - Successful [Tested: 21]
10.50#Ex3 𝒲 ⁑ { 𝗂 n ( 1 ) ⁑ ( z ) , 𝗂 n ( 2 ) ⁑ ( z ) } = ( - 1 ) n + 1 ⁒ z - 2 Wronskian spherical-Bessel-I-1 𝑛 𝑧 spherical-Bessel-I-2 𝑛 𝑧 superscript 1 𝑛 1 superscript 𝑧 2 {\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z% \right),{\mathsf{i}^{(2)}_{n}}\left(z\right)\right\}=(-1)^{n+1}z^{-2}}}
\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesseli{2}{n}@{z}} = (-1)^{n+1}z^{-2}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}} 
Error

Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n], Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n]}, z] == (- 1)^(n + 1)* (z)^(- 2)
Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[-0.5000000000000001, 0.8660254037844386]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.5000000000000001, -0.8660254037844386]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.50#Ex4 𝒲 ⁑ { 𝗂 n ( 1 ) ⁑ ( z ) , 𝗄 n ⁑ ( z ) } = 𝒲 ⁑ { 𝗂 n ( 2 ) ⁑ ( z ) , 𝗄 n ⁑ ( z ) } Wronskian spherical-Bessel-I-1 𝑛 𝑧 spherical-Bessel-K 𝑛 𝑧 Wronskian spherical-Bessel-I-2 𝑛 𝑧 spherical-Bessel-K 𝑛 𝑧 {\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z% \right),\mathsf{k}_{n}\left(z\right)\right\}=\mathscr{W}\left\{{\mathsf{i}^{(2% )}_{n}}\left(z\right),\mathsf{k}_{n}\left(z\right)\right\}\\ }}
\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesselK{n}@{z}} = \Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}} 
Error

Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z] == Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z]
Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[0.5384915109869794, 1.7026856201657974]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.6544302063904848, -2.4451654315616667]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.50#Ex4 𝒲 ⁑ { 𝗂 n ( 2 ) ⁑ ( z ) , 𝗄 n ⁑ ( z ) } = - 1 2 ⁒ Ο€ ⁒ z - 2 Wronskian spherical-Bessel-I-2 𝑛 𝑧 spherical-Bessel-K 𝑛 𝑧 1 2 πœ‹ superscript 𝑧 2 {\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(2)}_{n}}\left(z% \right),\mathsf{k}_{n}\left(z\right)\right\}\\ =-\tfrac{1}{2}\pi z^{-2}}}
\Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\ = -\tfrac{1}{2}\pi z^{-2}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}} 
Error

Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z] == -Divide[1,2]*Pi*(z)^(- 2)
Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[0.5161524079039588, -2.211692333258562]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[7.686727830477982, 4.996906619076774]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.50#Ex5 𝗃 n + 1 ⁑ ( z ) ⁒ 𝗒 n ⁑ ( z ) - 𝗃 n ⁑ ( z ) ⁒ 𝗒 n + 1 ⁑ ( z ) = z - 2 spherical-Bessel-J 𝑛 1 𝑧 spherical-Bessel-Y 𝑛 𝑧 spherical-Bessel-J 𝑛 𝑧 spherical-Bessel-Y 𝑛 1 𝑧 superscript 𝑧 2 {\displaystyle{\displaystyle\mathsf{j}_{n+1}\left(z\right)\mathsf{y}_{n}\left(% z\right)-\mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+1}\left(z\right)=z^{-2}}}
\sphBesselJ{n+1}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+1}@{z} = z^{-2}		 β„œ ⁑ ( ( ( n + 1 ) + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 ) - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - ( n + 1 ) - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( ( n + 1 ) + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(((n+1)+\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})% +k+1)>0,\Re((-(n+1)-\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-(n% +1)-\frac{1}{2}))+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))% +k+1)>0,\Re((-((n+1)+\frac{1}{2}))+k+1)>0}} 
Error

SphericalBesselJ[n + 1, z]*SphericalBesselY[n, z]- SphericalBesselJ[n, z]*SphericalBesselY[n + 1, z] == (z)^(- 2)
Missing Macro Error Successful - Successful [Tested: 21]
10.50#Ex6 𝗃 n + 2 ⁑ ( z ) ⁒ 𝗒 n ⁑ ( z ) - 𝗃 n ⁑ ( z ) ⁒ 𝗒 n + 2 ⁑ ( z ) = ( 2 ⁒ n + 3 ) ⁒ z - 3 spherical-Bessel-J 𝑛 2 𝑧 spherical-Bessel-Y 𝑛 𝑧 spherical-Bessel-J 𝑛 𝑧 spherical-Bessel-Y 𝑛 2 𝑧 2 𝑛 3 superscript 𝑧 3 {\displaystyle{\displaystyle\mathsf{j}_{n+2}\left(z\right)\mathsf{y}_{n}\left(% z\right)-\mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+2}\left(z\right)=(2n+3)z^{-% 3}}}
\sphBesselJ{n+2}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+2}@{z} = (2n+3)z^{-3}		 β„œ ⁑ ( ( ( n + 2 ) + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 2 ) - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - ( n + 2 ) - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( ( n + 2 ) + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 2 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 2 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 2 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 2 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(((n+2)+\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})% +k+1)>0,\Re((-(n+2)-\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-(n% +2)-\frac{1}{2}))+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))% +k+1)>0,\Re((-((n+2)+\frac{1}{2}))+k+1)>0}} 
Error

SphericalBesselJ[n + 2, z]*SphericalBesselY[n, z]- SphericalBesselJ[n, z]*SphericalBesselY[n + 2, z] == (2*n + 3)*(z)^(- 3)
Missing Macro Error Failure - Successful [Tested: 21]
10.50.E4 𝗃 0 ⁑ ( z ) ⁒ 𝗃 n ⁑ ( z ) + 𝗒 0 ⁑ ( z ) ⁒ 𝗒 n ⁑ ( z ) = cos ⁑ ( 1 2 ⁒ n ⁒ Ο€ ) ⁒ βˆ‘ k = 0 ⌊ n / 2 βŒ‹ ( - 1 ) k ⁒ a 2 ⁒ k ⁒ ( n + 1 2 ) z 2 ⁒ k + 2 + sin ⁑ ( 1 2 ⁒ n ⁒ Ο€ ) ⁒ βˆ‘ k = 0 ⌊ ( n - 1 ) / 2 βŒ‹ ( - 1 ) k ⁒ a 2 ⁒ k + 1 ⁒ ( n + 1 2 ) z 2 ⁒ k + 3 spherical-Bessel-J 0 𝑧 spherical-Bessel-J 𝑛 𝑧 spherical-Bessel-Y 0 𝑧 spherical-Bessel-Y 𝑛 𝑧 1 2 𝑛 πœ‹ superscript subscript π‘˜ 0 𝑛 2 superscript 1 π‘˜ subscript π‘Ž 2 π‘˜ 𝑛 1 2 superscript 𝑧 2 π‘˜ 2 1 2 𝑛 πœ‹ superscript subscript π‘˜ 0 𝑛 1 2 superscript 1 π‘˜ subscript π‘Ž 2 π‘˜ 1 𝑛 1 2 superscript 𝑧 2 π‘˜ 3 {\displaystyle{\displaystyle\mathsf{j}_{0}\left(z\right)\mathsf{j}_{n}\left(z% \right)+\mathsf{y}_{0}\left(z\right)\mathsf{y}_{n}\left(z\right)=\cos\left(% \tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac% {a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\sin\left(\tfrac{1}{2}n\pi\right)\sum_{k=0}% ^{\left\lfloor(n-1)/2\right\rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{% 2k+3}}}}
\sphBesselJ{0}@{z}\sphBesselJ{n}@{z}+\sphBesselY{0}@{z}\sphBesselY{n}@{z} = \cos@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\sin@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}}		 β„œ ⁑ ( ( 0 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 0 - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 0 - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 0 + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , k β‰₯ 1 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 π‘˜ 1 {\displaystyle{\displaystyle\Re((0+\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})+k+1% )>0,\Re((-0-\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-0-\frac{1}% {2}))+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(0+\frac{1}{2}))+k+1)>0,\Re(% (-(n+\frac{1}{2}))+k+1)>0,k\geq 1}} 
Error

SphericalBesselJ[0, z]*SphericalBesselJ[n, z]+ SphericalBesselY[0, z]*SphericalBesselY[n, z] == Cos[Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, Floor[n/2]}, GenerateConditions->None]+ Sin[Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k + 1]*(n +Divide[1,2]),(z)^(2*k + 3)], {k, 0, Floor[(n - 1)/2]}, GenerateConditions->None]
Missing Macro Error Failure - Skipped - Because timed out
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